Is there any field other than $\mathbb{F}_2$ which has $1+1=0$ ? My initial feeling was no but I don't know how to prove it. Can someone help ?

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    $\begingroup$ How about $\mathbb{F}_4$? In fact any field of characteristic $2$ would do it. $\endgroup$ – Stefan4024 Sep 8 '18 at 7:40

This property is called "characteristic $2$", and many fields have it. $\Bbb F_2$ is in some sense the most "basic" one, but we also have examples like $$ \Bbb F_4=\Bbb F_2[t]/(t^2+t+1) $$ or $\Bbb F_2(x)$, the field of rational functions in one variable with coefficients from $\Bbb F_2$.

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    $\begingroup$ "in some sense the most basic one" -- why not be precise and say: Every field of characteristic 2 contains a unique copy of $\Bbb F_2$, and conversely, every field which contains $\Bbb F_2$ is of characteristic 2. $\endgroup$ – Torsten Schoeneberg Sep 9 '18 at 23:58
  • $\begingroup$ Yeah, I could've done that. For some reason I didn't. And now you have. $\endgroup$ – Arthur Sep 10 '18 at 5:01

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